ICMCTA 2017: Coding Theory and Applications pp 163-191 | Cite as

Asymptotic Bounds for the Sizes of Constant Dimension Codes and an Improved Lower Bound

Conference paper
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10495)

Abstract

We study asymptotic lower and upper bounds for the sizes of constant dimension codes with respect to the subspace or injection distance, which is used in random linear network coding. In this context we review known upper bounds and show relations between them. A slightly improved version of the so-called linkage construction is presented which is e.g. used to construct constant dimension codes with subspace distance \(d=4\), dimension \(k=3\) of the codewords for all field sizes q, and sufficiently large dimensions v of the ambient space. It exceeds the MRD bound, for codes containing a lifted MRD code, by Etzion and Silberstein.

Keywords

Constant dimension codes Subspace distance Injection distance Random network coding 
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Notes

Acknowledgement

The authors would like to thank Harout Aydinian for providing an enlarged proof of Theorem 8, Natalia Silberstein for explaining the restriction \(3k \le v\) in [38, Corollary 39], Heide Gluesing-Luerssen for clarifying the independent origin of the linkage construction, and Alfred Wassermann for discussions about the asymptotic results of Frankl and Rödl. We thank the reviewers for their comments that helped us to improve the presentation of the paper.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.University of BayreuthBayreuthGermany

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